how can i convert a vertex of a plane into bezier with handle?
No, not the way you describe. You can convert a mesh to a collection of Straight splines using Alt+C in object mode, but the resulting Curve object doesn't have handles you can drag away from the Knot. (more about this later)

A simple mesh based solution is to use ctrl+shift+B with the vertex selected and drag out to generate the Bevel, or Fillet (to fillet use mouse wheel to set to number of cuts)



Regarding Spline / Bezier based 2d surfaces i recommend the answers to this question: Make a quarter curve with curves

You can convert the plane to a Curve object (using Alt+C as mentioned earlier, but this creates a Curve of type 'Poly', which means it doesn't have handles, only points. The curve object can be recast as a bezier type (but not easily from the UI, at least I'm not privy to any method). You can use the built in python console to convert the curve from 'POLY' to 'BEZIER' and it will create the handles in the right places too. Also remember to set the Curve type to 2D (from 3D) else you won't see the surfaces

&gt;&gt;&gt; # after you've converted the plane to a Curve object using Alt+C
&gt;&gt;&gt; obj = bpy.data.objects['Plane']
&gt;&gt;&gt; obj.data.splines[0].type
... 'POLY'
&gt;&gt;&gt; obj.data.splines[0].type = 'BEZIER'   # this sets it